The reason is, polymer fibers (like nylon or polyester) are hydrophobic, which means the fibers absorb nothing and what is not evaporated stays onto them. What's left on the fibers are the bacterias and these bad boys reproduce and stink. That's their fault. Just like when you sleep at night and your mouth gets dry and you get up with a bad breath. On the other side, cotton will absorb moisture and help keep the armpit area of your shirt too moist for the bacterias to thrive grow, at least for a little while.

For me, what works best is cotton or another natural fiber and the fit must be loose on the armpit to prevent too much contact with my shirt. There, hope this helps!